X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_1%2Fsty0%2Ffwd.ma;fp=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_1%2Fsty0%2Ffwd.ma;h=0000000000000000000000000000000000000000;hb=d2545ffd201b1aa49887313791386add78fa8603;hp=0d05ca147abaab635f5ebb329553b0d20ad08781;hpb=57ae1762497a5f3ea75740e2908e04adb8642cc2;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/basic_1/sty0/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/sty0/fwd.ma
deleted file mode 100644
index 0d05ca147..000000000
--- a/matita/matita/contribs/lambdadelta/basic_1/sty0/fwd.ma
+++ /dev/null
@@ -1,553 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/sty0/defs.ma".
-
-implied rec lemma sty0_ind (g: G) (P: (C \to (T \to (T \to Prop)))) (f: 
-(\forall (c: C).(\forall (n: nat).(P c (TSort n) (TSort (next g n)))))) (f0: 
-(\forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c 
-(CHead d (Bind Abbr) v)) \to (\forall (w: T).((sty0 g d v w) \to ((P d v w) 
-\to (P c (TLRef i) (lift (S i) O w))))))))))) (f1: (\forall (c: C).(\forall 
-(d: C).(\forall (v: T).(\forall (i: nat).((getl i c (CHead d (Bind Abst) v)) 
-\to (\forall (w: T).((sty0 g d v w) \to ((P d v w) \to (P c (TLRef i) (lift 
-(S i) O v))))))))))) (f2: (\forall (b: B).(\forall (c: C).(\forall (v: 
-T).(\forall (t1: T).(\forall (t2: T).((sty0 g (CHead c (Bind b) v) t1 t2) \to 
-((P (CHead c (Bind b) v) t1 t2) \to (P c (THead (Bind b) v t1) (THead (Bind 
-b) v t2)))))))))) (f3: (\forall (c: C).(\forall (v: T).(\forall (t1: 
-T).(\forall (t2: T).((sty0 g c t1 t2) \to ((P c t1 t2) \to (P c (THead (Flat 
-Appl) v t1) (THead (Flat Appl) v t2))))))))) (f4: (\forall (c: C).(\forall 
-(v1: T).(\forall (v2: T).((sty0 g c v1 v2) \to ((P c v1 v2) \to (\forall (t1: 
-T).(\forall (t2: T).((sty0 g c t1 t2) \to ((P c t1 t2) \to (P c (THead (Flat 
-Cast) v1 t1) (THead (Flat Cast) v2 t2)))))))))))) (c: C) (t: T) (t0: T) (s0: 
-sty0 g c t t0) on s0: P c t t0 \def match s0 with [(sty0_sort c0 n) 
-\Rightarrow (f c0 n) | (sty0_abbr c0 d v i g0 w s1) \Rightarrow (f0 c0 d v i 
-g0 w s1 ((sty0_ind g P f f0 f1 f2 f3 f4) d v w s1)) | (sty0_abst c0 d v i g0 
-w s1) \Rightarrow (f1 c0 d v i g0 w s1 ((sty0_ind g P f f0 f1 f2 f3 f4) d v w 
-s1)) | (sty0_bind b c0 v t1 t2 s1) \Rightarrow (f2 b c0 v t1 t2 s1 ((sty0_ind 
-g P f f0 f1 f2 f3 f4) (CHead c0 (Bind b) v) t1 t2 s1)) | (sty0_appl c0 v t1 
-t2 s1) \Rightarrow (f3 c0 v t1 t2 s1 ((sty0_ind g P f f0 f1 f2 f3 f4) c0 t1 
-t2 s1)) | (sty0_cast c0 v1 v2 s1 t1 t2 s2) \Rightarrow (f4 c0 v1 v2 s1 
-((sty0_ind g P f f0 f1 f2 f3 f4) c0 v1 v2 s1) t1 t2 s2 ((sty0_ind g P f f0 f1 
-f2 f3 f4) c0 t1 t2 s2))].
-
-lemma sty0_gen_sort:
- \forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((sty0 g c 
-(TSort n) x) \to (eq T x (TSort (next g n)))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda 
-(H: (sty0 g c (TSort n) x)).(insert_eq T (TSort n) (\lambda (t: T).(sty0 g c 
-t x)) (\lambda (_: T).(eq T x (TSort (next g n)))) (\lambda (y: T).(\lambda 
-(H0: (sty0 g c y x)).(sty0_ind g (\lambda (_: C).(\lambda (t: T).(\lambda 
-(t0: T).((eq T t (TSort n)) \to (eq T t0 (TSort (next g n))))))) (\lambda (_: 
-C).(\lambda (n0: nat).(\lambda (H1: (eq T (TSort n0) (TSort n))).(let H2 \def 
-(f_equal T nat (\lambda (e: T).(match e with [(TSort n1) \Rightarrow n1 | 
-(TLRef _) \Rightarrow n0 | (THead _ _ _) \Rightarrow n0])) (TSort n0) (TSort 
-n) H1) in (eq_ind_r nat n (\lambda (n1: nat).(eq T (TSort (next g n1)) (TSort 
-(next g n)))) (refl_equal T (TSort (next g n))) n0 H2))))) (\lambda (c0: 
-C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl i c0 
-(CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v 
-w)).(\lambda (_: (((eq T v (TSort n)) \to (eq T w (TSort (next g 
-n)))))).(\lambda (H4: (eq T (TLRef i) (TSort n))).(let H5 \def (eq_ind T 
-(TLRef i) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | 
-(TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (TSort n) 
-H4) in (False_ind (eq T (lift (S i) O w) (TSort (next g n))) H5))))))))))) 
-(\lambda (c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda 
-(_: (getl i c0 (CHead d (Bind Abst) v))).(\lambda (w: T).(\lambda (_: (sty0 g 
-d v w)).(\lambda (_: (((eq T v (TSort n)) \to (eq T w (TSort (next g 
-n)))))).(\lambda (H4: (eq T (TLRef i) (TSort n))).(let H5 \def (eq_ind T 
-(TLRef i) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | 
-(TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (TSort n) 
-H4) in (False_ind (eq T (lift (S i) O v) (TSort (next g n))) H5))))))))))) 
-(\lambda (b: B).(\lambda (c0: C).(\lambda (v: T).(\lambda (t1: T).(\lambda 
-(t2: T).(\lambda (_: (sty0 g (CHead c0 (Bind b) v) t1 t2)).(\lambda (_: (((eq 
-T t1 (TSort n)) \to (eq T t2 (TSort (next g n)))))).(\lambda (H3: (eq T 
-(THead (Bind b) v t1) (TSort n))).(let H4 \def (eq_ind T (THead (Bind b) v 
-t1) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H3) in 
-(False_ind (eq T (THead (Bind b) v t2) (TSort (next g n))) H4)))))))))) 
-(\lambda (c0: C).(\lambda (v: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
-(_: (sty0 g c0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 (TSort 
-(next g n)))))).(\lambda (H3: (eq T (THead (Flat Appl) v t1) (TSort n))).(let 
-H4 \def (eq_ind T (THead (Flat Appl) v t1) (\lambda (ee: T).(match ee with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) 
-\Rightarrow True])) I (TSort n) H3) in (False_ind (eq T (THead (Flat Appl) v 
-t2) (TSort (next g n))) H4))))))))) (\lambda (c0: C).(\lambda (v1: 
-T).(\lambda (v2: T).(\lambda (_: (sty0 g c0 v1 v2)).(\lambda (_: (((eq T v1 
-(TSort n)) \to (eq T v2 (TSort (next g n)))))).(\lambda (t1: T).(\lambda (t2: 
-T).(\lambda (_: (sty0 g c0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq 
-T t2 (TSort (next g n)))))).(\lambda (H5: (eq T (THead (Flat Cast) v1 t1) 
-(TSort n))).(let H6 \def (eq_ind T (THead (Flat Cast) v1 t1) (\lambda (ee: 
-T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
-| (THead _ _ _) \Rightarrow True])) I (TSort n) H5) in (False_ind (eq T 
-(THead (Flat Cast) v2 t2) (TSort (next g n))) H6)))))))))))) c y x H0))) 
-H))))).
-
-lemma sty0_gen_lref:
- \forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((sty0 g c 
-(TLRef n) x) \to (or (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda 
-(_: T).(getl n c (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: 
-T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(t: T).(eq T x (lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda 
-(u: T).(\lambda (_: T).(getl n c (CHead e (Bind Abst) u))))) (\lambda (e: 
-C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda 
-(u: T).(\lambda (_: T).(eq T x (lift (S n) O u)))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda 
-(H: (sty0 g c (TLRef n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(sty0 g c 
-t x)) (\lambda (_: T).(or (ex3_3 C T T (\lambda (e: C).(\lambda (u: 
-T).(\lambda (_: T).(getl n c (CHead e (Bind Abbr) u))))) (\lambda (e: 
-C).(\lambda (u: T).(\lambda (t0: T).(sty0 g e u t0)))) (\lambda (_: 
-C).(\lambda (_: T).(\lambda (t0: T).(eq T x (lift (S n) O t0)))))) (ex3_3 C T 
-T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead e (Bind 
-Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t0: T).(sty0 g e u 
-t0)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T x (lift (S n) O 
-u)))))))) (\lambda (y: T).(\lambda (H0: (sty0 g c y x)).(sty0_ind g (\lambda 
-(c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n)) \to (or (ex3_3 C 
-T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind 
-Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t1: T).(sty0 g e u 
-t1)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t1: T).(eq T t0 (lift (S n) 
-O t1)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl 
-n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda 
-(t1: T).(sty0 g e u t1)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: 
-T).(eq T t0 (lift (S n) O u))))))))))) (\lambda (c0: C).(\lambda (n0: 
-nat).(\lambda (H1: (eq T (TSort n0) (TLRef n))).(let H2 \def (eq_ind T (TSort 
-n0) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow True | (TLRef _) 
-\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef n) H1) in 
-(False_ind (or (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: 
-T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: 
-T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(t: T).(eq T (TSort (next g n0)) (lift (S n) O t)))))) (ex3_3 C T T (\lambda 
-(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) 
-(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda 
-(_: C).(\lambda (u: T).(\lambda (_: T).(eq T (TSort (next g n0)) (lift (S n) 
-O u))))))) H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda 
-(i: nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) v))).(\lambda (w: 
-T).(\lambda (H2: (sty0 g d v w)).(\lambda (_: (((eq T v (TLRef n)) \to (or 
-(ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n d (CHead 
-e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g 
-e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T w (lift (S n) 
-O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl 
-n d (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: 
-T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T w 
-(lift (S n) O u)))))))))).(\lambda (H4: (eq T (TLRef i) (TLRef n))).(let H5 
-\def (f_equal T nat (\lambda (e: T).(match e with [(TSort _) \Rightarrow i | 
-(TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef 
-n) H4) in (let H6 \def (eq_ind nat i (\lambda (n0: nat).(getl n0 c0 (CHead d 
-(Bind Abbr) v))) H1 n H5) in (eq_ind_r nat n (\lambda (n0: nat).(or (ex3_3 C 
-T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind 
-Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u 
-t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T (lift (S n0) O w) 
-(lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda 
-(_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: 
-T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda 
-(_: T).(eq T (lift (S n0) O w) (lift (S n) O u)))))))) (or_introl (ex3_3 C T 
-T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind 
-Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u 
-t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T (lift (S n) O w) 
-(lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda 
-(_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: 
-T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda 
-(_: T).(eq T (lift (S n) O w) (lift (S n) O u)))))) (ex3_3_intro C T T 
-(\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind 
-Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u 
-t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T (lift (S n) O w) 
-(lift (S n) O t))))) d v w H6 H2 (refl_equal T (lift (S n) O w)))) i 
-H5)))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: 
-nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abst) v))).(\lambda (w: 
-T).(\lambda (H2: (sty0 g d v w)).(\lambda (_: (((eq T v (TLRef n)) \to (or 
-(ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n d (CHead 
-e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g 
-e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T w (lift (S n) 
-O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl 
-n d (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: 
-T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T w 
-(lift (S n) O u)))))))))).(\lambda (H4: (eq T (TLRef i) (TLRef n))).(let H5 
-\def (f_equal T nat (\lambda (e: T).(match e with [(TSort _) \Rightarrow i | 
-(TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef 
-n) H4) in (let H6 \def (eq_ind nat i (\lambda (n0: nat).(getl n0 c0 (CHead d 
-(Bind Abst) v))) H1 n H5) in (eq_ind_r nat n (\lambda (n0: nat).(or (ex3_3 C 
-T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind 
-Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u 
-t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T (lift (S n0) O v) 
-(lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda 
-(_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: 
-T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda 
-(_: T).(eq T (lift (S n0) O v) (lift (S n) O u)))))))) (or_intror (ex3_3 C T 
-T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind 
-Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u 
-t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T (lift (S n) O v) 
-(lift (S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda 
-(_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: 
-T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda 
-(_: T).(eq T (lift (S n) O v) (lift (S n) O u)))))) (ex3_3_intro C T T 
-(\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind 
-Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u 
-t)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T (lift (S n) O v) 
-(lift (S n) O u))))) d v w H6 H2 (refl_equal T (lift (S n) O v)))) i 
-H5)))))))))))) (\lambda (b: B).(\lambda (c0: C).(\lambda (v: T).(\lambda (t1: 
-T).(\lambda (t2: T).(\lambda (_: (sty0 g (CHead c0 (Bind b) v) t1 
-t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (or (ex3_3 C T T (\lambda (e: 
-C).(\lambda (u: T).(\lambda (_: T).(getl n (CHead c0 (Bind b) v) (CHead e 
-(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e 
-u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(eq T t2 (lift (S n) 
-O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl 
-n (CHead c0 (Bind b) v) (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda 
-(u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: 
-T).(\lambda (_: T).(eq T t2 (lift (S n) O u)))))))))).(\lambda (H3: (eq T 
-(THead (Bind b) v t1) (TLRef n))).(let H4 \def (eq_ind T (THead (Bind b) v 
-t1) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H3) in 
-(False_ind (or (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: 
-T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: 
-T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (_: T).(\lambda 
-(t: T).(eq T (THead (Bind b) v t2) (lift (S n) O t)))))) (ex3_3 C T T 
-(\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind 
-Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u 
-t)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T (THead (Bind b) v 
-t2) (lift (S n) O u))))))) H4)))))))))) (\lambda (c0: C).(\lambda (v: 
-T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (sty0 g c0 t1 t2)).(\lambda 
-(_: (((eq T t1 (TLRef n)) \to (or (ex3_3 C T T (\lambda (e: C).(\lambda (u: 
-T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: 
-C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda 
-(_: T).(\lambda (t: T).(eq T t2 (lift (S n) O t)))))) (ex3_3 C T T (\lambda 
-(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) 
-(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda 
-(_: C).(\lambda (u: T).(\lambda (_: T).(eq T t2 (lift (S n) O 
-u)))))))))).(\lambda (H3: (eq T (THead (Flat Appl) v t1) (TLRef n))).(let H4 
-\def (eq_ind T (THead (Flat Appl) v t1) (\lambda (ee: T).(match ee with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) 
-\Rightarrow True])) I (TLRef n) H3) in (False_ind (or (ex3_3 C T T (\lambda 
-(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u))))) 
-(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (t: T).(eq T (THead (Flat Appl) v t2) (lift 
-(S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: 
-T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: 
-T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda 
-(_: T).(eq T (THead (Flat Appl) v t2) (lift (S n) O u))))))) H4))))))))) 
-(\lambda (c0: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (sty0 g c0 v1 
-v2)).(\lambda (_: (((eq T v1 (TLRef n)) \to (or (ex3_3 C T T (\lambda (e: 
-C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u))))) 
-(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (t: T).(eq T v2 (lift (S n) O t)))))) (ex3_3 
-C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e 
-(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e 
-u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T v2 (lift (S n) 
-O u)))))))))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (sty0 g c0 t1 
-t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (or (ex3_3 C T T (\lambda (e: 
-C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u))))) 
-(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (t: T).(eq T t2 (lift (S n) O t)))))) (ex3_3 
-C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e 
-(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e 
-u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq T t2 (lift (S n) 
-O u)))))))))).(\lambda (H5: (eq T (THead (Flat Cast) v1 t1) (TLRef n))).(let 
-H6 \def (eq_ind T (THead (Flat Cast) v1 t1) (\lambda (ee: T).(match ee with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) 
-\Rightarrow True])) I (TLRef n) H5) in (False_ind (or (ex3_3 C T T (\lambda 
-(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u))))) 
-(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(sty0 g e u t)))) (\lambda 
-(_: C).(\lambda (_: T).(\lambda (t: T).(eq T (THead (Flat Cast) v2 t2) (lift 
-(S n) O t)))))) (ex3_3 C T T (\lambda (e: C).(\lambda (u: T).(\lambda (_: 
-T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: 
-T).(\lambda (t: T).(sty0 g e u t)))) (\lambda (_: C).(\lambda (u: T).(\lambda 
-(_: T).(eq T (THead (Flat Cast) v2 t2) (lift (S n) O u))))))) H6)))))))))))) 
-c y x H0))) H))))).
-
-lemma sty0_gen_bind:
- \forall (g: G).(\forall (b: B).(\forall (c: C).(\forall (u: T).(\forall (t1: 
-T).(\forall (x: T).((sty0 g c (THead (Bind b) u t1) x) \to (ex2 T (\lambda 
-(t2: T).(sty0 g (CHead c (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T x (THead 
-(Bind b) u t2))))))))))
-\def
- \lambda (g: G).(\lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: 
-T).(\lambda (x: T).(\lambda (H: (sty0 g c (THead (Bind b) u t1) 
-x)).(insert_eq T (THead (Bind b) u t1) (\lambda (t: T).(sty0 g c t x)) 
-(\lambda (_: T).(ex2 T (\lambda (t2: T).(sty0 g (CHead c (Bind b) u) t1 t2)) 
-(\lambda (t2: T).(eq T x (THead (Bind b) u t2))))) (\lambda (y: T).(\lambda 
-(H0: (sty0 g c y x)).(sty0_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda 
-(t0: T).((eq T t (THead (Bind b) u t1)) \to (ex2 T (\lambda (t2: T).(sty0 g 
-(CHead c0 (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T t0 (THead (Bind b) u 
-t2)))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (H1: (eq T (TSort n) 
-(THead (Bind b) u t1))).(let H2 \def (eq_ind T (TSort n) (\lambda (ee: 
-T).(match ee with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | 
-(THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t1) H1) in (False_ind 
-(ex2 T (\lambda (t2: T).(sty0 g (CHead c0 (Bind b) u) t1 t2)) (\lambda (t2: 
-T).(eq T (TSort (next g n)) (THead (Bind b) u t2)))) H2))))) (\lambda (c0: 
-C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl i c0 
-(CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v 
-w)).(\lambda (_: (((eq T v (THead (Bind b) u t1)) \to (ex2 T (\lambda (t2: 
-T).(sty0 g (CHead d (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T w (THead (Bind 
-b) u t2))))))).(\lambda (H4: (eq T (TLRef i) (THead (Bind b) u t1))).(let H5 
-\def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow 
-False])) I (THead (Bind b) u t1) H4) in (False_ind (ex2 T (\lambda (t2: 
-T).(sty0 g (CHead c0 (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T (lift (S i) O 
-w) (THead (Bind b) u t2)))) H5))))))))))) (\lambda (c0: C).(\lambda (d: 
-C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind 
-Abst) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v w)).(\lambda (_: (((eq T 
-v (THead (Bind b) u t1)) \to (ex2 T (\lambda (t2: T).(sty0 g (CHead d (Bind 
-b) u) t1 t2)) (\lambda (t2: T).(eq T w (THead (Bind b) u t2))))))).(\lambda 
-(H4: (eq T (TLRef i) (THead (Bind b) u t1))).(let H5 \def (eq_ind T (TLRef i) 
-(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t1) 
-H4) in (False_ind (ex2 T (\lambda (t2: T).(sty0 g (CHead c0 (Bind b) u) t1 
-t2)) (\lambda (t2: T).(eq T (lift (S i) O v) (THead (Bind b) u t2)))) 
-H5))))))))))) (\lambda (b0: B).(\lambda (c0: C).(\lambda (v: T).(\lambda (t0: 
-T).(\lambda (t2: T).(\lambda (H1: (sty0 g (CHead c0 (Bind b0) v) t0 
-t2)).(\lambda (H2: (((eq T t0 (THead (Bind b) u t1)) \to (ex2 T (\lambda (t3: 
-T).(sty0 g (CHead (CHead c0 (Bind b0) v) (Bind b) u) t1 t3)) (\lambda (t3: 
-T).(eq T t2 (THead (Bind b) u t3))))))).(\lambda (H3: (eq T (THead (Bind b0) 
-v t0) (THead (Bind b) u t1))).(let H4 \def (f_equal T B (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow b0 | (TLRef _) \Rightarrow b0 | 
-(THead k _ _) \Rightarrow (match k with [(Bind b1) \Rightarrow b1 | (Flat _) 
-\Rightarrow b0])])) (THead (Bind b0) v t0) (THead (Bind b) u t1) H3) in ((let 
-H5 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow v | 
-(TLRef _) \Rightarrow v | (THead _ t _) \Rightarrow t])) (THead (Bind b0) v 
-t0) (THead (Bind b) u t1) H3) in ((let H6 \def (f_equal T T (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | 
-(THead _ _ t) \Rightarrow t])) (THead (Bind b0) v t0) (THead (Bind b) u t1) 
-H3) in (\lambda (H7: (eq T v u)).(\lambda (H8: (eq B b0 b)).(let H9 \def 
-(eq_ind T t0 (\lambda (t: T).((eq T t (THead (Bind b) u t1)) \to (ex2 T 
-(\lambda (t3: T).(sty0 g (CHead (CHead c0 (Bind b0) v) (Bind b) u) t1 t3)) 
-(\lambda (t3: T).(eq T t2 (THead (Bind b) u t3)))))) H2 t1 H6) in (let H10 
-\def (eq_ind T t0 (\lambda (t: T).(sty0 g (CHead c0 (Bind b0) v) t t2)) H1 t1 
-H6) in (let H11 \def (eq_ind T v (\lambda (t: T).((eq T t1 (THead (Bind b) u 
-t1)) \to (ex2 T (\lambda (t3: T).(sty0 g (CHead (CHead c0 (Bind b0) t) (Bind 
-b) u) t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3)))))) H9 u H7) 
-in (let H12 \def (eq_ind T v (\lambda (t: T).(sty0 g (CHead c0 (Bind b0) t) 
-t1 t2)) H10 u H7) in (eq_ind_r T u (\lambda (t: T).(ex2 T (\lambda (t3: 
-T).(sty0 g (CHead c0 (Bind b) u) t1 t3)) (\lambda (t3: T).(eq T (THead (Bind 
-b0) t t2) (THead (Bind b) u t3))))) (let H13 \def (eq_ind B b0 (\lambda (b1: 
-B).((eq T t1 (THead (Bind b) u t1)) \to (ex2 T (\lambda (t3: T).(sty0 g 
-(CHead (CHead c0 (Bind b1) u) (Bind b) u) t1 t3)) (\lambda (t3: T).(eq T t2 
-(THead (Bind b) u t3)))))) H11 b H8) in (let H14 \def (eq_ind B b0 (\lambda 
-(b1: B).(sty0 g (CHead c0 (Bind b1) u) t1 t2)) H12 b H8) in (eq_ind_r B b 
-(\lambda (b1: B).(ex2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind b) u) t1 
-t3)) (\lambda (t3: T).(eq T (THead (Bind b1) u t2) (THead (Bind b) u t3))))) 
-(ex_intro2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind b) u) t1 t3)) (\lambda 
-(t3: T).(eq T (THead (Bind b) u t2) (THead (Bind b) u t3))) t2 H14 
-(refl_equal T (THead (Bind b) u t2))) b0 H8))) v H7)))))))) H5)) H4)))))))))) 
-(\lambda (c0: C).(\lambda (v: T).(\lambda (t0: T).(\lambda (t2: T).(\lambda 
-(_: (sty0 g c0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind b) u t1)) \to 
-(ex2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind b) u) t1 t3)) (\lambda (t3: 
-T).(eq T t2 (THead (Bind b) u t3))))))).(\lambda (H3: (eq T (THead (Flat 
-Appl) v t0) (THead (Bind b) u t1))).(let H4 \def (eq_ind T (THead (Flat Appl) 
-v t0) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef 
-_) \Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _) 
-\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u t1) 
-H3) in (False_ind (ex2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind b) u) t1 
-t3)) (\lambda (t3: T).(eq T (THead (Flat Appl) v t2) (THead (Bind b) u t3)))) 
-H4))))))))) (\lambda (c0: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: 
-(sty0 g c0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind b) u t1)) \to (ex2 T 
-(\lambda (t2: T).(sty0 g (CHead c0 (Bind b) u) t1 t2)) (\lambda (t2: T).(eq T 
-v2 (THead (Bind b) u t2))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: 
-(sty0 g c0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind b) u t1)) \to (ex2 T 
-(\lambda (t3: T).(sty0 g (CHead c0 (Bind b) u) t1 t3)) (\lambda (t3: T).(eq T 
-t2 (THead (Bind b) u t3))))))).(\lambda (H5: (eq T (THead (Flat Cast) v1 t0) 
-(THead (Bind b) u t1))).(let H6 \def (eq_ind T (THead (Flat Cast) v1 t0) 
-(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _) 
-\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u t1) 
-H5) in (False_ind (ex2 T (\lambda (t3: T).(sty0 g (CHead c0 (Bind b) u) t1 
-t3)) (\lambda (t3: T).(eq T (THead (Flat Cast) v2 t2) (THead (Bind b) u 
-t3)))) H6)))))))))))) c y x H0))) H))))))).
-
-lemma sty0_gen_appl:
- \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (x: 
-T).((sty0 g c (THead (Flat Appl) u t1) x) \to (ex2 T (\lambda (t2: T).(sty0 g 
-c t1 t2)) (\lambda (t2: T).(eq T x (THead (Flat Appl) u t2)))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (x: 
-T).(\lambda (H: (sty0 g c (THead (Flat Appl) u t1) x)).(insert_eq T (THead 
-(Flat Appl) u t1) (\lambda (t: T).(sty0 g c t x)) (\lambda (_: T).(ex2 T 
-(\lambda (t2: T).(sty0 g c t1 t2)) (\lambda (t2: T).(eq T x (THead (Flat 
-Appl) u t2))))) (\lambda (y: T).(\lambda (H0: (sty0 g c y x)).(sty0_ind g 
-(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Flat Appl) 
-u t1)) \to (ex2 T (\lambda (t2: T).(sty0 g c0 t1 t2)) (\lambda (t2: T).(eq T 
-t0 (THead (Flat Appl) u t2)))))))) (\lambda (c0: C).(\lambda (n: 
-nat).(\lambda (H1: (eq T (TSort n) (THead (Flat Appl) u t1))).(let H2 \def 
-(eq_ind T (TSort n) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow 
-True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I 
-(THead (Flat Appl) u t1) H1) in (False_ind (ex2 T (\lambda (t2: T).(sty0 g c0 
-t1 t2)) (\lambda (t2: T).(eq T (TSort (next g n)) (THead (Flat Appl) u t2)))) 
-H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: 
-nat).(\lambda (_: (getl i c0 (CHead d (Bind Abbr) v))).(\lambda (w: 
-T).(\lambda (_: (sty0 g d v w)).(\lambda (_: (((eq T v (THead (Flat Appl) u 
-t1)) \to (ex2 T (\lambda (t2: T).(sty0 g d t1 t2)) (\lambda (t2: T).(eq T w 
-(THead (Flat Appl) u t2))))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat 
-Appl) u t1))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee 
-with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ 
-_) \Rightarrow False])) I (THead (Flat Appl) u t1) H4) in (False_ind (ex2 T 
-(\lambda (t2: T).(sty0 g c0 t1 t2)) (\lambda (t2: T).(eq T (lift (S i) O w) 
-(THead (Flat Appl) u t2)))) H5))))))))))) (\lambda (c0: C).(\lambda (d: 
-C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind 
-Abst) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v w)).(\lambda (_: (((eq T 
-v (THead (Flat Appl) u t1)) \to (ex2 T (\lambda (t2: T).(sty0 g d t1 t2)) 
-(\lambda (t2: T).(eq T w (THead (Flat Appl) u t2))))))).(\lambda (H4: (eq T 
-(TLRef i) (THead (Flat Appl) u t1))).(let H5 \def (eq_ind T (TLRef i) 
-(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) u 
-t1) H4) in (False_ind (ex2 T (\lambda (t2: T).(sty0 g c0 t1 t2)) (\lambda 
-(t2: T).(eq T (lift (S i) O v) (THead (Flat Appl) u t2)))) H5))))))))))) 
-(\lambda (b: B).(\lambda (c0: C).(\lambda (v: T).(\lambda (t0: T).(\lambda 
-(t2: T).(\lambda (_: (sty0 g (CHead c0 (Bind b) v) t0 t2)).(\lambda (_: (((eq 
-T t0 (THead (Flat Appl) u t1)) \to (ex2 T (\lambda (t3: T).(sty0 g (CHead c0 
-(Bind b) v) t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Flat Appl) u 
-t3))))))).(\lambda (H3: (eq T (THead (Bind b) v t0) (THead (Flat Appl) u 
-t1))).(let H4 \def (eq_ind T (THead (Bind b) v t0) (\lambda (ee: T).(match ee 
-with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ 
-_) \Rightarrow (match k with [(Bind _) \Rightarrow True | (Flat _) 
-\Rightarrow False])])) I (THead (Flat Appl) u t1) H3) in (False_ind (ex2 T 
-(\lambda (t3: T).(sty0 g c0 t1 t3)) (\lambda (t3: T).(eq T (THead (Bind b) v 
-t2) (THead (Flat Appl) u t3)))) H4)))))))))) (\lambda (c0: C).(\lambda (v: 
-T).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H1: (sty0 g c0 t0 
-t2)).(\lambda (H2: (((eq T t0 (THead (Flat Appl) u t1)) \to (ex2 T (\lambda 
-(t3: T).(sty0 g c0 t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Flat Appl) u 
-t3))))))).(\lambda (H3: (eq T (THead (Flat Appl) v t0) (THead (Flat Appl) u 
-t1))).(let H4 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
-\Rightarrow v | (TLRef _) \Rightarrow v | (THead _ t _) \Rightarrow t])) 
-(THead (Flat Appl) v t0) (THead (Flat Appl) u t1) H3) in ((let H5 \def 
-(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t0 | (TLRef 
-_) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Flat Appl) v t0) 
-(THead (Flat Appl) u t1) H3) in (\lambda (H6: (eq T v u)).(let H7 \def 
-(eq_ind T t0 (\lambda (t: T).((eq T t (THead (Flat Appl) u t1)) \to (ex2 T 
-(\lambda (t3: T).(sty0 g c0 t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Flat 
-Appl) u t3)))))) H2 t1 H5) in (let H8 \def (eq_ind T t0 (\lambda (t: T).(sty0 
-g c0 t t2)) H1 t1 H5) in (eq_ind_r T u (\lambda (t: T).(ex2 T (\lambda (t3: 
-T).(sty0 g c0 t1 t3)) (\lambda (t3: T).(eq T (THead (Flat Appl) t t2) (THead 
-(Flat Appl) u t3))))) (ex_intro2 T (\lambda (t3: T).(sty0 g c0 t1 t3)) 
-(\lambda (t3: T).(eq T (THead (Flat Appl) u t2) (THead (Flat Appl) u t3))) t2 
-H8 (refl_equal T (THead (Flat Appl) u t2))) v H6))))) H4))))))))) (\lambda 
-(c0: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (sty0 g c0 v1 
-v2)).(\lambda (_: (((eq T v1 (THead (Flat Appl) u t1)) \to (ex2 T (\lambda 
-(t2: T).(sty0 g c0 t1 t2)) (\lambda (t2: T).(eq T v2 (THead (Flat Appl) u 
-t2))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (sty0 g c0 t0 
-t2)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u t1)) \to (ex2 T (\lambda 
-(t3: T).(sty0 g c0 t1 t3)) (\lambda (t3: T).(eq T t2 (THead (Flat Appl) u 
-t3))))))).(\lambda (H5: (eq T (THead (Flat Cast) v1 t0) (THead (Flat Appl) u 
-t1))).(let H6 \def (eq_ind T (THead (Flat Cast) v1 t0) (\lambda (ee: 
-T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
-| (THead k _ _) \Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat 
-f) \Rightarrow (match f with [Appl \Rightarrow False | Cast \Rightarrow 
-True])])])) I (THead (Flat Appl) u t1) H5) in (False_ind (ex2 T (\lambda (t3: 
-T).(sty0 g c0 t1 t3)) (\lambda (t3: T).(eq T (THead (Flat Cast) v2 t2) (THead 
-(Flat Appl) u t3)))) H6)))))))))))) c y x H0))) H)))))).
-
-lemma sty0_gen_cast:
- \forall (g: G).(\forall (c: C).(\forall (v1: T).(\forall (t1: T).(\forall 
-(x: T).((sty0 g c (THead (Flat Cast) v1 t1) x) \to (ex3_2 T T (\lambda (v2: 
-T).(\lambda (_: T).(sty0 g c v1 v2))) (\lambda (_: T).(\lambda (t2: T).(sty0 
-g c t1 t2))) (\lambda (v2: T).(\lambda (t2: T).(eq T x (THead (Flat Cast) v2 
-t2))))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (v1: T).(\lambda (t1: T).(\lambda 
-(x: T).(\lambda (H: (sty0 g c (THead (Flat Cast) v1 t1) x)).(insert_eq T 
-(THead (Flat Cast) v1 t1) (\lambda (t: T).(sty0 g c t x)) (\lambda (_: 
-T).(ex3_2 T T (\lambda (v2: T).(\lambda (_: T).(sty0 g c v1 v2))) (\lambda 
-(_: T).(\lambda (t2: T).(sty0 g c t1 t2))) (\lambda (v2: T).(\lambda (t2: 
-T).(eq T x (THead (Flat Cast) v2 t2)))))) (\lambda (y: T).(\lambda (H0: (sty0 
-g c y x)).(sty0_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq 
-T t (THead (Flat Cast) v1 t1)) \to (ex3_2 T T (\lambda (v2: T).(\lambda (_: 
-T).(sty0 g c0 v1 v2))) (\lambda (_: T).(\lambda (t2: T).(sty0 g c0 t1 t2))) 
-(\lambda (v2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Cast) v2 t2))))))))) 
-(\lambda (c0: C).(\lambda (n: nat).(\lambda (H1: (eq T (TSort n) (THead (Flat 
-Cast) v1 t1))).(let H2 \def (eq_ind T (TSort n) (\lambda (ee: T).(match ee 
-with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ 
-_) \Rightarrow False])) I (THead (Flat Cast) v1 t1) H1) in (False_ind (ex3_2 
-T T (\lambda (v2: T).(\lambda (_: T).(sty0 g c0 v1 v2))) (\lambda (_: 
-T).(\lambda (t2: T).(sty0 g c0 t1 t2))) (\lambda (v2: T).(\lambda (t2: T).(eq 
-T (TSort (next g n)) (THead (Flat Cast) v2 t2))))) H2))))) (\lambda (c0: 
-C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl i c0 
-(CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v 
-w)).(\lambda (_: (((eq T v (THead (Flat Cast) v1 t1)) \to (ex3_2 T T (\lambda 
-(v2: T).(\lambda (_: T).(sty0 g d v1 v2))) (\lambda (_: T).(\lambda (t2: 
-T).(sty0 g d t1 t2))) (\lambda (v2: T).(\lambda (t2: T).(eq T w (THead (Flat 
-Cast) v2 t2)))))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat Cast) v1 
-t1))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee with 
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) 
-\Rightarrow False])) I (THead (Flat Cast) v1 t1) H4) in (False_ind (ex3_2 T T 
-(\lambda (v2: T).(\lambda (_: T).(sty0 g c0 v1 v2))) (\lambda (_: T).(\lambda 
-(t2: T).(sty0 g c0 t1 t2))) (\lambda (v2: T).(\lambda (t2: T).(eq T (lift (S 
-i) O w) (THead (Flat Cast) v2 t2))))) H5))))))))))) (\lambda (c0: C).(\lambda 
-(d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d 
-(Bind Abst) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v w)).(\lambda (_: 
-(((eq T v (THead (Flat Cast) v1 t1)) \to (ex3_2 T T (\lambda (v2: T).(\lambda 
-(_: T).(sty0 g d v1 v2))) (\lambda (_: T).(\lambda (t2: T).(sty0 g d t1 t2))) 
-(\lambda (v2: T).(\lambda (t2: T).(eq T w (THead (Flat Cast) v2 
-t2)))))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat Cast) v1 t1))).(let H5 
-\def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow 
-False])) I (THead (Flat Cast) v1 t1) H4) in (False_ind (ex3_2 T T (\lambda 
-(v2: T).(\lambda (_: T).(sty0 g c0 v1 v2))) (\lambda (_: T).(\lambda (t2: 
-T).(sty0 g c0 t1 t2))) (\lambda (v2: T).(\lambda (t2: T).(eq T (lift (S i) O 
-v) (THead (Flat Cast) v2 t2))))) H5))))))))))) (\lambda (b: B).(\lambda (c0: 
-C).(\lambda (v: T).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (sty0 g 
-(CHead c0 (Bind b) v) t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) v1 
-t1)) \to (ex3_2 T T (\lambda (v2: T).(\lambda (_: T).(sty0 g (CHead c0 (Bind 
-b) v) v1 v2))) (\lambda (_: T).(\lambda (t3: T).(sty0 g (CHead c0 (Bind b) v) 
-t1 t3))) (\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) v2 
-t3)))))))).(\lambda (H3: (eq T (THead (Bind b) v t0) (THead (Flat Cast) v1 
-t1))).(let H4 \def (eq_ind T (THead (Bind b) v t0) (\lambda (ee: T).(match ee 
-with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ 
-_) \Rightarrow (match k with [(Bind _) \Rightarrow True | (Flat _) 
-\Rightarrow False])])) I (THead (Flat Cast) v1 t1) H3) in (False_ind (ex3_2 T 
-T (\lambda (v2: T).(\lambda (_: T).(sty0 g c0 v1 v2))) (\lambda (_: 
-T).(\lambda (t3: T).(sty0 g c0 t1 t3))) (\lambda (v2: T).(\lambda (t3: T).(eq 
-T (THead (Bind b) v t2) (THead (Flat Cast) v2 t3))))) H4)))))))))) (\lambda 
-(c0: C).(\lambda (v: T).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (sty0 
-g c0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) v1 t1)) \to (ex3_2 T 
-T (\lambda (v2: T).(\lambda (_: T).(sty0 g c0 v1 v2))) (\lambda (_: 
-T).(\lambda (t3: T).(sty0 g c0 t1 t3))) (\lambda (v2: T).(\lambda (t3: T).(eq 
-T t2 (THead (Flat Cast) v2 t3)))))))).(\lambda (H3: (eq T (THead (Flat Appl) 
-v t0) (THead (Flat Cast) v1 t1))).(let H4 \def (eq_ind T (THead (Flat Appl) v 
-t0) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _) 
-\Rightarrow False | (Flat f) \Rightarrow (match f with [Appl \Rightarrow True 
-| Cast \Rightarrow False])])])) I (THead (Flat Cast) v1 t1) H3) in (False_ind 
-(ex3_2 T T (\lambda (v2: T).(\lambda (_: T).(sty0 g c0 v1 v2))) (\lambda (_: 
-T).(\lambda (t3: T).(sty0 g c0 t1 t3))) (\lambda (v2: T).(\lambda (t3: T).(eq 
-T (THead (Flat Appl) v t2) (THead (Flat Cast) v2 t3))))) H4))))))))) (\lambda 
-(c0: C).(\lambda (v0: T).(\lambda (v2: T).(\lambda (H1: (sty0 g c0 v0 
-v2)).(\lambda (H2: (((eq T v0 (THead (Flat Cast) v1 t1)) \to (ex3_2 T T 
-(\lambda (v3: T).(\lambda (_: T).(sty0 g c0 v1 v3))) (\lambda (_: T).(\lambda 
-(t2: T).(sty0 g c0 t1 t2))) (\lambda (v3: T).(\lambda (t2: T).(eq T v2 (THead 
-(Flat Cast) v3 t2)))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H3: 
-(sty0 g c0 t0 t2)).(\lambda (H4: (((eq T t0 (THead (Flat Cast) v1 t1)) \to 
-(ex3_2 T T (\lambda (v3: T).(\lambda (_: T).(sty0 g c0 v1 v3))) (\lambda (_: 
-T).(\lambda (t3: T).(sty0 g c0 t1 t3))) (\lambda (v3: T).(\lambda (t3: T).(eq 
-T t2 (THead (Flat Cast) v3 t3)))))))).(\lambda (H5: (eq T (THead (Flat Cast) 
-v0 t0) (THead (Flat Cast) v1 t1))).(let H6 \def (f_equal T T (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow v0 | (TLRef _) \Rightarrow v0 | 
-(THead _ t _) \Rightarrow t])) (THead (Flat Cast) v0 t0) (THead (Flat Cast) 
-v1 t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e with 
-[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) 
-\Rightarrow t])) (THead (Flat Cast) v0 t0) (THead (Flat Cast) v1 t1) H5) in 
-(\lambda (H8: (eq T v0 v1)).(let H9 \def (eq_ind T t0 (\lambda (t: T).((eq T 
-t (THead (Flat Cast) v1 t1)) \to (ex3_2 T T (\lambda (v3: T).(\lambda (_: 
-T).(sty0 g c0 v1 v3))) (\lambda (_: T).(\lambda (t3: T).(sty0 g c0 t1 t3))) 
-(\lambda (v3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) v3 t3))))))) H4 
-t1 H7) in (let H10 \def (eq_ind T t0 (\lambda (t: T).(sty0 g c0 t t2)) H3 t1 
-H7) in (let H11 \def (eq_ind T v0 (\lambda (t: T).((eq T t (THead (Flat Cast) 
-v1 t1)) \to (ex3_2 T T (\lambda (v3: T).(\lambda (_: T).(sty0 g c0 v1 v3))) 
-(\lambda (_: T).(\lambda (t3: T).(sty0 g c0 t1 t3))) (\lambda (v3: 
-T).(\lambda (t3: T).(eq T v2 (THead (Flat Cast) v3 t3))))))) H2 v1 H8) in 
-(let H12 \def (eq_ind T v0 (\lambda (t: T).(sty0 g c0 t v2)) H1 v1 H8) in 
-(ex3_2_intro T T (\lambda (v3: T).(\lambda (_: T).(sty0 g c0 v1 v3))) 
-(\lambda (_: T).(\lambda (t3: T).(sty0 g c0 t1 t3))) (\lambda (v3: 
-T).(\lambda (t3: T).(eq T (THead (Flat Cast) v2 t2) (THead (Flat Cast) v3 
-t3)))) v2 t2 H12 H10 (refl_equal T (THead (Flat Cast) v2 t2))))))))) 
-H6)))))))))))) c y x H0))) H)))))).
-